Related papers: The Zrank Conjecture and Restricted Cauchy Matrice…

We study random partitions $\lambda=(\lambda_1,\lambda_2,...,\lambda_d)$ of $n$ whose length is not bigger than a fixed number $d$. Suppose a random partition $\lambda$ is distributed according to the Jack measure, which is a deformation of…

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is…

The purpose of this note is to give an insertion scheme proof of the formula, $$p_\mu = \sum_{\lambda\vdash k} \chi^\lambda(\mu)s_\lambda,\formula$$ where $p_\mu$ is the power sum symmetric function, $s_\lambda$ is the Schur function and…

We use a double-duality argument to give a new proof of Dieudonn\'e's theorem on spaces of singular matrices. The argument connects the situation to the structure of spaces of operators with rank at most $1$, and works best over…

The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting…

We classify the $Q$-homogeneous skew Schur $Q$-functions, i.e., those of the form $Q_{\lambda/\mu} = k \cdot Q_{\nu}$. On the way we develop new tools that are useful also in the context of other classification problems for skew Schur…

We study the asymptotics of Schur polynomials with partitions $\lambda$ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots,(m-1),0)$ by at most one component at the beginning as…

Using techniques developed in \cite{KLR}, we verify Sarnak's conjecture for two classes of rank-one subshifts with unbounded cutting parameters. The first class of rank-one subshifts we consider are called {\em almost complete congruency…

In this paper, using crystal theory we prove the existence of a new family of irreducible components appearing in the tensor product of two irreducible integrable highest weight modules over symmetrizable Kac-Moody algebras motivated by the…

We develop a topological framework for proving lower bounds on sign-rank via $\mathbb{Z}_2$-equivariant topology, and use it to resolve the sign-rank of the Gap Hamming Distance problem up to lower-order terms. For every (partial) sign…

Let the dimension $N$ of data and the sample size $T$ tend to $\infty$ with $N/T \to c > 0$. The spectral properties of a sample correlation matrix $\mathbf{C}$ and a sample covariance matrix $\mathbf{S}$ are asymptotically equal whenever…

This article provides new lower bounds for both Schur and weak Schur numbers by exploiting a "template"-based approach. The concept of "template" is also generalized to weak Schur numbers. Finding new templates leads to explicit partitions…

A partition statistic ` crank' gives combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of…

In this paper, we discuss several additional properties a power linear Keller map may have. The Structural Conjecture by Druzkowski in [Dru] asserts that two such properties are equivalent, but we show that one of this properties is…

The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam…

Following the breakthrough of Croot, Lev, and Pach, Tao introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which…

We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of…

We develop an elementary theory of divisibility on the monoid $M(n,R)^\times$ consisting of all square matrices of size $n\ge 1$ of non-zero determinants with coefficients in a principal ideal domain $R$. In particular, we show that any…

We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free…

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We…